Theorem ipasslem1 9831

Description: Lemma for ipassi 9842. Show the inner product associative law for nonnegative integers.

Hypotheses

Ref	Expression
ip1i.1	|- X = (BaseSet` U)
ip1i.2	|- G = (+v` U)
ip1i.4	|- S = (.s` U)
ip1i.7	|- P = (.i` U)
ip1i.9	|- U e. CPreHil
ipasslem1.b	|- B e. X

Assertion

Ref	Expression
ipasslem1	|- ((N e. NN0 /\ A e. X) -> ((NSA)PB) = (N x. (APB)))

Proof of Theorem ipasslem1

Step	Hyp	Ref	Expression
1		ip1i.9	. . . . . . . . . . . 12 |- U e. CPreHil
2	1	phnvi 9816	. . . . . . . . . . 11 |- U e. NrmCVec
3		ip1i.1	. . . . . . . . . . . 12 |- X = (BaseSet` U)
4		ip1i.4	. . . . . . . . . . . 12 |- S = (.s` U)
5	3, 4	nvscl 9579	. . . . . . . . . . 11 |- ((U e. NrmCVec /\ k e. CC /\ A e. X) -> (kSA) e. X)
6	2, 5	mp3an1 1178	. . . . . . . . . 10 |- ((k e. CC /\ A e. X) -> (kSA) e. X)
7		nn0cn 7318	. . . . . . . . . 10 |- (k e. NN0 -> k e. CC)
8	6, 7	sylan 497	. . . . . . . . 9 |- ((k e. NN0 /\ A e. X) -> (kSA) e. X)
9		ipasslem1.b	. . . . . . . . . 10 |- B e. X
10		ip1i.2	. . . . . . . . . . 11 |- G = (+v` U)
11		ip1i.7	. . . . . . . . . . 11 |- P = (.i` U)
12	3, 10, 4, 11, 1	ipdiri 9830	. . . . . . . . . 10 |- (((kSA) e. X /\ A e. X /\ B e. X) -> (((kSA)GA)PB) = (((kSA)PB) + (APB)))
13	9, 12	mp3an3 1180	. . . . . . . . 9 |- (((kSA) e. X /\ A e. X) -> (((kSA)GA)PB) = (((kSA)PB) + (APB)))
14	8, 13	sylancom 531	. . . . . . . 8 |- ((k e. NN0 /\ A e. X) -> (((kSA)GA)PB) = (((kSA)PB) + (APB)))
15		ax1cn 6422	. . . . . . . . . . . 12 |- 1 e. CC
16	3, 10, 4	nvdir 9584	. . . . . . . . . . . . 13 |- ((U e. NrmCVec /\ (k e. CC /\ 1 e. CC /\ A e. X)) -> ((k + 1)SA) = ((kSA)G(1SA)))
17	2, 16	mpan 759	. . . . . . . . . . . 12 |- ((k e. CC /\ 1 e. CC /\ A e. X) -> ((k + 1)SA) = ((kSA)G(1SA)))
18	15, 17	mp3an2 1179	. . . . . . . . . . 11 |- ((k e. CC /\ A e. X) -> ((k + 1)SA) = ((kSA)G(1SA)))
19	18, 7	sylan 497	. . . . . . . . . 10 |- ((k e. NN0 /\ A e. X) -> ((k + 1)SA) = ((kSA)G(1SA)))
20	3, 4	nvsid 9580	. . . . . . . . . . . . 13 |- ((U e. NrmCVec /\ A e. X) -> (1SA) = A)
21	2, 20	mpan 759	. . . . . . . . . . . 12 |- (A e. X -> (1SA) = A)
22	21	adantl 424	. . . . . . . . . . 11 |- ((k e. NN0 /\ A e. X) -> (1SA) = A)
23	22	opreq2d 4898	. . . . . . . . . 10 |- ((k e. NN0 /\ A e. X) -> ((kSA)G(1SA)) = ((kSA)GA))
24	19, 23	eqtrd 1925	. . . . . . . . 9 |- ((k e. NN0 /\ A e. X) -> ((k + 1)SA) = ((kSA)GA))
25	24	opreq1d 4897	. . . . . . . 8 |- ((k e. NN0 /\ A e. X) -> (((k + 1)SA)PB) = (((kSA)GA)PB))
26	3, 11	ipcl 9704	. . . . . . . . . . . 12 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (APB) e. CC)
27	2, 9, 26	mp3an13 1182	. . . . . . . . . . 11 |- (A e. X -> (APB) e. CC)
28		mulid2 6578	. . . . . . . . . . 11 |- ((APB) e. CC -> (1 x. (APB)) = (APB))
29	27, 28	syl 12	. . . . . . . . . 10 |- (A e. X -> (1 x. (APB)) = (APB))
30	29	adantl 424	. . . . . . . . 9 |- ((k e. NN0 /\ A e. X) -> (1 x. (APB)) = (APB))
31	30	opreq2d 4898	. . . . . . . 8 |- ((k e. NN0 /\ A e. X) -> (((kSA)PB) + (1 x. (APB))) = (((kSA)PB) + (APB)))
32	14, 25, 31	3eqtr4d 1937	. . . . . . 7 |- ((k e. NN0 /\ A e. X) -> (((k + 1)SA)PB) = (((kSA)PB) + (1 x. (APB))))
33		opreq1 4889	. . . . . . 7 |- (((kSA)PB) = (k x. (APB)) -> (((kSA)PB) + (1 x. (APB))) = ((k x. (APB)) + (1 x. (APB))))
34	32, 33	sylan9eq 1948	. . . . . 6 |- (((k e. NN0 /\ A e. X) /\ ((kSA)PB) = (k x. (APB))) -> (((k + 1)SA)PB) = ((k x. (APB)) + (1 x. (APB))))
35		adddir 6472	. . . . . . . . 9 |- ((k e. CC /\ 1 e. CC /\ (APB) e. CC) -> ((k + 1) x. (APB)) = ((k x. (APB)) + (1 x. (APB))))
36	15, 35	mp3an2 1179	. . . . . . . 8 |- ((k e. CC /\ (APB) e. CC) -> ((k + 1) x. (APB)) = ((k x. (APB)) + (1 x. (APB))))
37	36, 7, 27	syl2an 503	. . . . . . 7 |- ((k e. NN0 /\ A e. X) -> ((k + 1) x. (APB)) = ((k x. (APB)) + (1 x. (APB))))
38	37	adantr 425	. . . . . 6 |- (((k e. NN0 /\ A e. X) /\ ((kSA)PB) = (k x. (APB))) -> ((k + 1) x. (APB)) = ((k x. (APB)) + (1 x. (APB))))
39	34, 38	eqtr4d 1928	. . . . 5 |- (((k e. NN0 /\ A e. X) /\ ((kSA)PB) = (k x. (APB))) -> (((k + 1)SA)PB) = ((k + 1) x. (APB)))
40	39	exp31 407	. . . 4 |- (k e. NN0 -> (A e. X -> (((kSA)PB) = (k x. (APB)) -> (((k + 1)SA)PB) = ((k + 1) x. (APB)))))
41	40	a2d 16	. . 3 |- (k e. NN0 -> ((A e. X -> ((kSA)PB) = (k x. (APB))) -> (A e. X -> (((k + 1)SA)PB) = ((k + 1) x. (APB)))))
42		eqid 1884	. . . . . 6 |- (0v` U) = (0v` U)
43	3, 42, 11	ip0l 9710	. . . . 5 |- ((U e. NrmCVec /\ B e. X) -> ((0v` U)PB) = 0)
44	2, 9, 43	mp2an 761	. . . 4 |- ((0v` U)PB) = 0
45	3, 4, 42	nv0 9590	. . . . . 6 |- ((U e. NrmCVec /\ A e. X) -> (0SA) = (0v` U))
46	2, 45	mpan 759	. . . . 5 |- (A e. X -> (0SA) = (0v` U))
47	46	opreq1d 4897	. . . 4 |- (A e. X -> ((0SA)PB) = ((0v` U)PB))
48		mul02 6607	. . . . 5 |- ((APB) e. CC -> (0 x. (APB)) = 0)
49	27, 48	syl 12	. . . 4 |- (A e. X -> (0 x. (APB)) = 0)
50	44, 47, 49	3eqtr4a 1954	. . 3 |- (A e. X -> ((0SA)PB) = (0 x. (APB)))
51		opreq1 4889	. . . . . 6 |- (j = 0 -> (jSA) = (0SA))
52	51	opreq1d 4897	. . . . 5 |- (j = 0 -> ((jSA)PB) = ((0SA)PB))
53		opreq1 4889	. . . . 5 |- (j = 0 -> (j x. (APB)) = (0 x. (APB)))
54	52, 53	eqeq12d 1899	. . . 4 |- (j = 0 -> (((jSA)PB) = (j x. (APB)) <-> ((0SA)PB) = (0 x. (APB))))
55	54	imbi2d 674	. . 3 |- (j = 0 -> ((A e. X -> ((jSA)PB) = (j x. (APB))) <-> (A e. X -> ((0SA)PB) = (0 x. (APB)))))
56		opreq1 4889	. . . . . 6 |- (j = k -> (jSA) = (kSA))
57	56	opreq1d 4897	. . . . 5 |- (j = k -> ((jSA)PB) = ((kSA)PB))
58		opreq1 4889	. . . . 5 |- (j = k -> (j x. (APB)) = (k x. (APB)))
59	57, 58	eqeq12d 1899	. . . 4 |- (j = k -> (((jSA)PB) = (j x. (APB)) <-> ((kSA)PB) = (k x. (APB))))
60	59	imbi2d 674	. . 3 |- (j = k -> ((A e. X -> ((jSA)PB) = (j x. (APB))) <-> (A e. X -> ((kSA)PB) = (k x. (APB)))))
61		opreq1 4889	. . . . . 6 |- (j = (k + 1) -> (jSA) = ((k + 1)SA))
62	61	opreq1d 4897	. . . . 5 |- (j = (k + 1) -> ((jSA)PB) = (((k + 1)SA)PB))
63		opreq1 4889	. . . . 5 |- (j = (k + 1) -> (j x. (APB)) = ((k + 1) x. (APB)))
64	62, 63	eqeq12d 1899	. . . 4 |- (j = (k + 1) -> (((jSA)PB) = (j x. (APB)) <-> (((k + 1)SA)PB) = ((k + 1) x. (APB))))
65	64	imbi2d 674	. . 3 |- (j = (k + 1) -> ((A e. X -> ((jSA)PB) = (j x. (APB))) <-> (A e. X -> (((k + 1)SA)PB) = ((k + 1) x. (APB)))))
66		opreq1 4889	. . . . . 6 |- (j = N -> (jSA) = (NSA))
67	66	opreq1d 4897	. . . . 5 |- (j = N -> ((jSA)PB) = ((NSA)PB))
68		opreq1 4889	. . . . 5 |- (j = N -> (j x. (APB)) = (N x. (APB)))
69	67, 68	eqeq12d 1899	. . . 4 |- (j = N -> (((jSA)PB) = (j x. (APB)) <-> ((NSA)PB) = (N x. (APB))))
70	69	imbi2d 674	. . 3 |- (j = N -> ((A e. X -> ((jSA)PB) = (j x. (APB))) <-> (A e. X -> ((NSA)PB) = (N x. (APB)))))
71	41, 50, 55, 60, 65, 70	nn0indALT 7425	. 2 |- (N e. NN0 -> (A e. X -> ((NSA)PB) = (N x. (APB))))
72	71	imp 377	1 |- ((N e. NN0 /\ A e. X) -> ((NSA)PB) = (N x. (APB)))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  ` cfv 3998  (class class class)co 4884  CCcc 6384  0cc0 6386  1c1 6387   + caddc 6389   x. cmul 6391  NN0cn0 6450  NrmCVeccnv 9535  +vcpv 9536  BaseSetcba 9537  .scns 9538  0vcn0v 9539  .icip 9688  CPreHilcphl 9812

This theorem is referenced by:  ipasslem2 9832  ipasslem3 9833  ipasslem4 9834

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-3 7155  df-4 7156  df-n0 7309  df-z 7345  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-sum 8240  df-grp 9316  df-gid 9317  df-ginv 9318  df-abl 9408  df-vc 9497  df-nv 9543  df-va 9546  df-ba 9547  df-sm 9548  df-0v 9549  df-nm 9551  df-ip 9689  df-ph 9813

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